Given two graphs $G$ and $H$, the {it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of whose edges have different colors. Denote by $kK_2$ a matching of size $k$ and $mathcal {T}_n$ the class of all plane triangulations of order $n$, respectively. In [S. Jendrol$$, I. Schiermeyer and J. Tu, Rainbow numbers for matchings in plane triangulations, Discrete Math. 331(2014), 158--164], the authors determined the exact values of $rb(mathcal {T}_n, kK_2)$ for $2leq k le 4$ and proved that $2n+2k-9 le rb(mathcal {T}_n, kK_2) le 2n+2k-7+2binom{2k-2}{3}$ for $k ge 5$. In this paper, we improve the upper bounds and prove that $rb(mathcal {T}_n, kK_2)le 2n+6k-16$ for $n ge 2k$ and $kge 5$. Especially, we show that $rb(mathcal {T}_n, 5K_2)=2n+1$ for $n ge 11$.
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